
<TITLE>prob007: all-interval series</TITLE>
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<H1>prob007: all-interval series</H1>

<TABLE>
<TR> <TD> proposed by
     <TD ALIGN=LEFT> <A HREF="http://www.cs.ubc.ca/spider/hoos/">
          <B>Holger Hoos</B></A> 
          <ADDRESS><a href="mailto:hoos@cs.ubc.ca">
          hoos@cs.ubc.ca</a></ADDRESS>
</TABLE>
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<H3> Specification </H3>

<TT>
Given the twelve standard pitch-classes (c, c#, d, ...), represented
by numbers 0,1,...,11, find a series in which each
pitch-class occurs exactly once and in which the musical
intervals between neighbouring notes cover the full set of
intervals from the minor second (1 semitone) to the
major seventh (11 semitones). That is, for each of the intervals,
there is a pair of neigbhouring pitch-classes in the series,
between which this interval appears.

The problem of finding such a series can be easily
formulated as an instance of a more general arithmetic problem on
Z_n, the set of integer residues modulo n.
Given n in N, find a vector s = (s_1, ..., s_n), such that
(i) s is a permutation of Z_n = {0,1,...,n-1};
and (ii) the interval vector
v = (|s_2-s_1|, |s_3-s_2|, ... |s_n-s_{n-1}|)
is a permutation of Z_n-{0} = {1,2,...,n-1}.
A vector v satisfying these conditions is called
an all-interval series of size n;
the problem of finding such a series is the
all-interval series problem of size n.
We may also be interested in finding all possible
series of a given size.

</TT>

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